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# Notational Conventions

This section provides the notational conventions used throughout this thesis.

A digital image on a 2D finite arithmetic lattice is a function that maps the supporting lattice onto a finite set of signal values (e.g., grey levels, colours, or multi-band signatures). For simplicity, the set is assumed to be integer-valued in the range between 0 and .

A pixel in an image is denoted by where is the positioning index in the lattice . The set of all pixels in an image with exception of a pixel is denoted by . Similarly, all sites on lattice except the th site is denoted by , so that .

The notation defines a neighbourhood of a site , which usually consists of a set of sites with a specific spatial configuration. All sites in a neighbourhood are dependent of each other.

Because the thesis deals mainly with probability texture models, in particular Markov-Gibbs random fields, related terminology and notations are presented below.

A random variable has a nondeterministic value with a given probability distribution. A sequence or array of statistically interrelated random variables form a discrete stochastic process. Let a random variable be associated with each site . Then these variables form a random field  in a configuration space  if a probability measure exists. A random field is a two-dimensional stochastic process. The configuration space has a combinatorial number of different images, .

Under a random field model, an image is considered as an instance or a sample of the configuration space . The image probability , or simply, , is known as the joint probability of the image signals. Another commonly-used probability measure is conditional probability of a pixel given all other pixels , denoted by . The conditional probability is a local probability of image signals, while the joint probability is a global one.

For convenience's sake, the notational conventions and symbols used in this thesis are summarised in Table 4.1.

 Notation Meaning A 2D finite arithmetic lattice The set of integral signal values A digital image defined on All pixels in the image except the pixel A random field The configuration space of a random field Neighbourhood of the site The lattice with exception of the th site All sites of with exception of a site Joint probability of a sample image Conditional probability of image signal at pixel given

Next: Texture Analysis and Synthesis: Up: Introduction Previous: Organisation of the Thesis
dzho002 2006-02-22